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Properties

Label 2-390-13.3-c1-0-6
Degree $2$
Conductor $390$
Sign $0.859 + 0.511i$
Analytic cond. $3.11416$
Root an. cond. $1.76469$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s − 5-s + (0.499 − 0.866i)6-s + (2.5 − 4.33i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (1.5 + 2.59i)11-s + 0.999·12-s + (−2.5 − 2.59i)13-s + 5·14-s + (0.5 + 0.866i)15-s + (−0.5 − 0.866i)16-s + (4 − 6.92i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s − 0.447·5-s + (0.204 − 0.353i)6-s + (0.944 − 1.63i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.158 − 0.273i)10-s + (0.452 + 0.783i)11-s + 0.288·12-s + (−0.693 − 0.720i)13-s + 1.33·14-s + (0.129 + 0.223i)15-s + (−0.125 − 0.216i)16-s + (0.970 − 1.68i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.859 + 0.511i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.859 + 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37206 - 0.377090i\)
\(L(\frac12)\) \(\approx\) \(1.37206 - 0.377090i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + T \)
13 \( 1 + (2.5 + 2.59i)T \)
good7 \( 1 + (-2.5 + 4.33i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-4 + 6.92i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.5 + 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2 - 3.46i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 + (-3.5 - 6.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3 + 5.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3 - 5.19i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3T + 47T^{2} \)
53 \( 1 - T + 53T^{2} \)
59 \( 1 + (6 - 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4 + 6.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1 - 1.73i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 + 2T + 79T^{2} \)
83 \( 1 - 8T + 83T^{2} \)
89 \( 1 + (-5.5 - 9.52i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.49609631662168158332695502945, −10.41513631541863398691472256687, −9.383577281595628268269351009975, −7.88127892647357733272462422502, −7.37674679339913567309724890579, −6.91521316024092169532631252059, −5.12793525022361651353586049475, −4.66754358556719115491956771686, −3.20538254383081894545876644311, −0.982624931644370404091049194855, 1.80402139087837853590352056605, 3.28864928362008128579029329385, 4.43515130368606390189860790858, 5.47887409253854378312717854064, 6.17842300625523799904966333156, 8.017329282219072338940324245706, 8.711213230850735519341622482761, 9.655925829411925947797474293582, 10.71019029806201543855336521509, 11.58781777065658408086541356225

Graph of the $Z$-function along the critical line